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is there a relationship

if so, what is the equation

use the equation for prediction

Overviewexists between two variables when one of them is related to the other in some way

Definition1. The sample of paired data (x,y) is a random sample.

2. The pairs of (x,y) data have a bivariate normal distribution.

AssumptionsScatterplot (or scatter diagram)

is a graph in which the paired (x,y) sample data are plotted with a horizontal x axis and a vertical y axis. Each individual (x,y) pair is plotted as a single point.

DefinitionPositive Linear Correlation

y

y

y

x

x

x

(a) Positive

(b) Strong

positive

(c) Perfect

positive

Scatter Plots

Negative Linear Correlation

y

y

y

x

x

x

(d) Negative

(e) Strong

negative

(f) Perfect

negative

Scatter Plots

xy/n - (x/n)(y/n)

r =

(SDx) (SDy)

- Definition
- Linear Correlation Coefficient r

measures strength of the linear relationship between paired x and y values in a sample

Where xy/n is the mean of the cross products; (x/n) is the mean of the x variable; (y/n) is the mean of the y variable; SDx is the standard deviation of the x variable and SDy is the standard deviation of the x variable

n number of pairs of data presented

denotes the addition of the items indicated.

x/ndenotes the mean of all x values.

y/ndenotes the mean of all y values.

xy/ndenotes the mean of the cross products [x times y, summed; divided by n]

r linear correlation coefficient for a sample

linear correlation coefficient for a population

Notation for the Linear Correlation CoefficientRound to three decimal places

Use calculator or computer if possible

Rounding the Linear Correlation Coefficient r

1. -1 r 1

2. Value of r does not change if all values of either variable are converted to a different scale.

3. The r is not affected by the choice of x and y. Interchange x and y and the value of r will not change.

4. r measures strength of a linear relationship.

Properties of the Linear Correlation Coefficient rIf the absolute value of r exceeds the value in Sig. Table, conclude that there is a significant linear correlation.

Otherwise, there is not sufficient evidence to support the conclusion of significant linear correlation.

Remember to use n-2

Interpreting the Linear Correlation Coefficient1. Causation: It is wrong to conclude that correlation implies causality.

2. Averages: Averages suppress individual variation and may inflate the correlation coefficient.

3. Linearity: There may be some relationship between x and y even when there is no significant linear correlation.

Common Errors Involving Correlation200

150

Distance

(feet)

100

50

0

1

2

3

4

5

6

7

8

0

Time (seconds)

Common Errors Involving CorrelationCorrelation Calculations

Rank Order Correlation -RhoPearson’s - r

Rank Order Correlation, cont

Rho = 1- [6 (∑D2) / N (N2-1)]

Rho = 1- [6(58)/10(102-1)]

Rho = 1- [348 / 10 (100 -1)]

Rho = 1- [348 / 990]

Rho = 1- 0.352

Rho = 0.648

(∑D2 = 58)

N=10

xy/n - (x/n)(y/n)

r =

(SDx) (SDy)

Pearson’s rr = 32.86 - (5.5) (5.5)/(3.03) (3.03)

r = 35.86 - 30.25 / 9.09

r = 5.61 / 9.09

r = 0.6172

Pearson’s r

Excel Demonstration

x Plastic (lb)

0.27

2

1.41

3

2.19

3

2.83

6

2.19

4

1.81

2

0.85

1

3.05

5

y Household

Is there a significant linear correlation?

x Plastic (lb)

0.27

2

1.41

3

2.19

3

2.83

6

2.19

4

1.81

2

0.85

1

3.05

5

y Household

Is there a significant linear correlation?

x Plastic (lb)

0.27

2

1.41

3

2.19

3

2.83

6

2.19

4

1.81

2

0.85

1

3.05

5

y Household

r = 0.842

R2 = 0.71

Is there a significant linear correlation?

= .01

= .05

n

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

25

30

35

40

45

50

60

70

80

90

100

.950

.878

.811

.754

.707

.666

.632

.602

.576

.553

.532

.514

.497

.482

.468

.456

.444

.396

.361

.335

.312

.294

.279

.254

.236

.220

.207

.196

.999

.959

.917

.875

.834

.798

.765

.735

.708

.684

.661

.641

.623

.606

.590

.575

.561

.505

.463

.430

.402

.378

.361

.330

.305

.286

.269

.256

Is there a significant linear correlation?

n = 8 = 0.05 H0: = 0

H1 : 0

Test statistic is r = 0.842

Critical values are r = - 0.707 and 0.707

(Table R with n = 8 and = 0.05)

TABLE R Critical Values of the Pearson Correlation Coefficient r

Is there a significant linear correlation?

0.842 > 0.707, That is the test statistic does fall within the

critical region.

Therefore, we REJECT H0: = 0 (no correlation) and conclude

there is a significant linear correlation between the weights of

discarded plastic and household size.

Reject

= 0

Fail to reject

= 0

Reject

= 0

1

- 1

r = - 0.707

0

r = 0.707

Sample data:

r = 0.842

(follows format of earlier chapters)

To determine whether there is a significant linear correlation between two variables

Two methods

Both methods let H0: =

(no significant linear correlation)

H1:

(significant linear correlation)

Formal Hypothesis TestTest statistic: correlation between two variablesr

Critical values: Refer to Table R

(no degrees of freedom)

Method 2: Test Statistic is r

(uses fewer calculations)

Test statistic: correlation between two variablesr

Critical values: Refer to Table A-6

(no degrees of freedom)

Method 2: Test Statistic is r

(uses fewer calculations)

Reject

= 0

Fail to reject

= 0

Reject

= 0

r = 0.811

1

r = - 0.811

0

-1

Sample data:

r = 0.828

Method 1: Test Statistic is correlation between two variablest

(follows format of earlier chapters)

Test statistic:

r

t =

1 - r 2

n - 2

Critical values:

use Table T with

degrees of freedom = n - 2

Select a correlation between two variables

significance

level

Calculate r using

Formula 9-1

r

The test statistic is

The test statistic is

r

t =

Critical values of t are from

Table A-6

1 - r 2

n -2

Critical values of t are from Table A-3 with n-2 degrees of freedom

If the absolute value of the

test statistic exceeds the

critical values, reject H0: = 0

Otherwise fail to reject H0

If H0 is rejected conclude that there

is a significant linear correlation.

If you fail to reject H0, then there is

not sufficient evidence to conclude

that there is linear correlation.

Start

Let H0: = 0

H1: 0

METHOD 1

METHOD 2

Testing for a

Linear Correlation

Why does the critical value of correlation between two variablesr increase as sample size decreases?

A correlation by chance is more likely.

Coefficient of Determination correlation between two variables(Effect Size)

r2

The part of variance of one variable that can be explained by the variance of a related variable.

Justification for correlation between two variablesr Formula

(x -x) (y -y)

r=

(x, y) centroid of sample points

(n -1) Sx Sy

x = 3

y

x - x = 7- 3 = 4

(7, 23)

•

24

20

y - y = 23 - 11 = 12

Quadrant 1

Quadrant 2

16

•

12

y = 11

(x, y)

•

8

Quadrant 3

Quadrant 4

•

•

4

x

0

0

1

2

3

4

5

6

7

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